1. Identify the given elements and their properties:
- $\triangle ABC$ is a right-angled triangle at $A$.
- $D$ is the foot of the altitude from $A$ to $BC$.
- $O_1$ and $O_2$ are the incenters of $\triangle ADB$ and $\triangle ADC$, respectively.
- The circle with center $A$ and radius $AD$ intersects $AB$ at $K$ and $AC$ at $L$.

2. **Analyze the circle with center $A$ and radius $AD$:**
- Since $AD$ is the altitude, $AD$ is perpendicular to $BC$.
- The circle intersects $AB$ at $K$ and $AC$ at $L$, implying $AK = AD = AL$.

3. Consider the angles formed by the circle and the triangle:
- Since $\angle A$ is a right angle, $\angle B + \angle C = 90^\circ$.
- The circle with center $A$ and radius $AD$ implies that $\angle DAL = \angle DAK = 90^\circ$.

4. Analyze the tangent properties and angles:
- Since $CD$ is a tangent to the circle at $D$, $\angle CDL = \frac{1}{2} \angle DAL = \frac{1}{2} \angle B$.
- Given $\angle CDL + \angle LDO_2 = 45^\circ$, we have $\angle LDO_2 = \frac{1}{2} \angle C = \angle LCO_2$.

5. **Prove that points $C, L, O_2, D$ are concyclic:**
- Since $\angle LDO_2 = \angle LCO_2$, quadrilateral $CLOD$ is cyclic.
- Therefore, $\angle ALO_2 = \angle CDO_2 = 45^\circ$.

6. **Show that $O_2$ lies on line $KL$:**
- Since $\angle ALO_2 = 45^\circ = \angle ALK$, it follows that $O_2 \in KL$.

7. **Similarly, show that $O_1$ lies on line $KL$:**
- By similar arguments, $O_1 \in KL$.

8. **Conclude that $O_1, O_2, K, L$ are collinear:**
- Since both $O_1$ and $O_2$ lie on line $KL$, the points $O_1, O_2, K, L$ are collinear.

$\blacksquare$